Tham khảo tài liệu 'stewart - calculus - early transcendentals 6e hq (thomson, 2008) episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | II INFINITE SEQUENCES AND SERIES The partial sums Tn of a Taylor series provide better and better approximations to a function as n increases. Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zeno s paradoxes and the decimal representation of numbers. Their importance in calculus stems from Newton s idea of representing functions as sums of infinite series. For instance in finding areas he often integrated a function by first expressing it as a series and then integrating each term of the series. We will pursue his idea in Section in order to integrate such functions as e x 2. Recall that we have previously been unable to do this. Many of the functions that arise in mathematical physics and chemistry such as Bessel functions are defined as sums of series so it is important to be familiar with the basic concepts of convergence of infinite sequences and series. Physicists also use series in another way as we will see in Section . In studying fields as diverse as optics special relativity and electromagnetism they analyze phenomena by replacing a function with the first few terms in the series that represents it. 674 I SEQUENCES A sequence can be thought of as a list of numbers written in a definite order a1 a2 a3 a4 . an . The number a1 is called the first term a2 is the second term and in general an is the nth term. We will deal exclusively with infinite sequences and so each term an will have a successor an 1 . Notice that for every positive integer n there is a corresponding number an and so a sequence can be defined as a function whose domain is the set of positive integers. But we usually write an instead of the function notation f n for the value of the function at the number n . I NOTATION The sequence ai a2 a-3 . is also denoted by an or an n i EXAMPLE I Some sequences can be defined by giving a formula for the nth term. In the following examples we give three descriptions of the sequence one by
